Physics · 18
Attempt these before you open a book or a video. Write the argument in the work box and the conclusion in the answer box. Check counts the problem only when both pass, and it does not open the solution. The solution stays locked for twenty minutes after you start the timer.
Attack 1
Show that \(y=A\sin(kx-\omega t)\) travels in the positive \(x\) direction with speed \(\omega/k\), if \(\omega\) and \(k\) are positive.
Timer not started. Twenty minutes of real work before a solution.
A crest is where the phase is \(\pi/2\), so \(kx-\omega t=\pi/2\), hence \(x=\omega t/k+\mathrm{const}\). The crest moves at \(\omega/k\) toward increasing \(x\).
Attack 2
Two slits separated by \(d\) are illuminated by wavelength \(\lambda\). Where are the bright fringes on a distant screen?
Timer not started. Twenty minutes of real work before a solution.
The path difference is \(d\sin\theta\). Bright fringes when that is an integer number of wavelengths: \(d\sin\theta=m\lambda\). On a distant screen, \(y\approx L\tan\theta\approx L\sin\theta\) for small angles, so \(y_m\approx m\lambda L/d\).
Attack 3
A standing wave \(2A\sin(kx)\cos(\omega t)\) has nodes where?
Timer not started. Twenty minutes of real work before a solution.
Where \(\sin(kx)=0\), so \(x=n\pi/k\), independent of time. Those points never move.
This page is oscillations that travel. A wave on a string, sound, and light are the same mathematics with different restoring mechanisms: a shape \(f(x-vt)\) that propagates, a linear wave equation, and superposition. Interference is what superposition does when two waves occupy the same place. Optics in this course means geometrical rays as the short-wavelength limit, and then interference and diffraction as the wave phenomena that rays cannot explain: double slit, single slit, and thin films.
OpenStax Volume 3 and MIT 8.03 are the working pair. The course is called vibrations and waves and it is the right level: normal modes, dispersion, and the wave equation, not a museum of lens formulas. Geometrical optics is in OpenStax and you should be able to do a thin lens, but it is not the ceiling. The ceiling is an 8.03 exam and the optics problems in Irodov that require a path difference.
A phase you cannot track is an interference problem you will get backwards. Write the phase before you write the intensity.
Statements only. Write the argument before you reveal it. A revealed proof you did not attempt is a paragraph you will not be able to rebuild at 2am.
The wave equation. \(u_{tt}=v^2 u_{xx}\) has right- and left-moving solutions \(f(x-vt)\) and \(g(x+vt)\). On a finite string with fixed ends, the normal modes are standing waves with a discrete spectrum.
The traveling-wave claim is the factoring you did for the PDE page, or a direct chain-rule check. Fixed ends force nodes at the endpoints, hence \(k=n\pi/L\) and frequencies that are integer multiples of the fundamental.
Superposition and beats. The sum of two close frequencies is a rapid oscillation inside a slow envelope. The beat frequency is the difference of the frequencies.
Use the sum-to-product identity from the precalculus page. The envelope is the slow cosine. Count how often the envelope peaks.
Impedance and reflection. At a boundary, part of a wave reflects and part transmits. A fixed end inverts the pulse. A free end does not. The reflection coefficient is fixed by continuity of the transverse displacement and the balance of transverse forces.
Write the incoming, reflected, and transmitted waves, match the displacement at the joint, and match the slopes with the tensions. The fixed-end limit is infinite mass on the far side, which forces the reflected amplitude to be minus the incoming amplitude.
Interference. Coherent sources interfere. The phase difference is \(2\pi\) times the path difference over \(\lambda\), plus any reflection phase shift of \(\pi\). Constructive interference when the phase difference is a multiple of \(2\pi\).
Superpose \(A\cos(\omega t)\) and \(A\cos(\omega t+\phi)\). The amplitude is \(2A\cos(\phi/2)\). Maxima when \(\phi=2\pi m\).
Diffraction and thin lenses. A single slit of width \(a\) has minima at \(a\sin\theta=m\lambda\) for nonzero integers \(m\). A thin lens in the paraxial limit obeys \(1/s+1/s'=1/f\). Ray optics is the limit in which interference fringes are too fine to resolve.
The single-slit minima come from pairing Huygens sources across the slit so each pair differs by half a wavelength. The lens equation is similar triangles in the paraxial ray diagram, or Fermat’s principle. State which one you are using.
Four rungs, in order. Do not skip a rung because the next one looks more serious.
Watch or read once. Do not study from this.
Feynman Lectures, the waves chapters via the official site
Read once, do not study from this. Volume I contains the waves and the start of optics. Read the contents and one chapter. Then go to 8.03.
The thing you work through.
MIT 8.03SC and OpenStax Volume 3
8.03 is the course. OpenStax Volume 3 holds the optics chapters in a standard order: https://openstax.org/details/books/university-physics-volume-3
The problem book.
8.03 problem sets
Normal modes, dispersion, and boundary conditions. This is the hard layer. Lens worksheets are not.
The hardest questions that exist at this level.
Irodov, oscillations and wave optics
The ceiling. Path differences, coherence, and waves on strings with boundary conditions.
Names and links. The problem text stays on the source site. Copyrighted contest problems are not copied here. What follows is where they are and why they hurt.
Pass: one 8.03 exam, plus written derivations of the traveling-wave speed, the fixed-end normal modes, and the double-slit maxima. Five Irodov optics or wave problems. Fail if a phase shift of \(\pi\) on reflection is inserted without a reason.
Five sets. Finish a set before you touch the next. Full sentences, not a circled answer.
Ten problems to meet again. Logging a writeup schedules them at 1 day, 7 days, and 30 days in this browser.
Derive the speed of a sinusoidal wave, state the fixed-end spectrum, and derive the double-slit maxima. No notes.